Acronym	gyquerco (alt.: rers quigybcu)
Name	inverto-elongated retrograde square gyrobicupola, gyrated quasirhombicuboctahedron, pseudo great rhombicuboctahedron, retroelongated retrosquare gyrobicupola
Circumradius	sqrt[5-2 sqrt(2)]/2 = 0.736813
	©
Dihedral angles	between {4} and {4}:   45° between {3} and {4}:   arccos(sqrt(2/3)) = 35.264390°
Face vector	24, 48, 26
Confer	uniform relative: stop   querco   non-convex cap: rasquacu   related Johnson solids: esquigybcu
External links

There also is a non-gyrated axial stack too: rasquacu + inv stop + ortho rasquacu = querco, one of the uniform solids. In fact, both have the same vertex figure all over. But querco features full cubical symmetry, while gyquerco features 4-fold antiprismatic symmetry only, thereby dividing its vertex set into 2 classes.

And just as querco allows for a non-quasi-version, sirco, this gyrated stack too will have an according non-quasi-version: squacu + op + gyro squacu = esquigybcu (J37).

Incidence matrix according to Dynkin symbol

xxxo4/3oxxx&#(x,-x)t   → height(1,2) = height(3,4) = 1/sqrt(2) = 0.707107
                         height(2,3) = -1
({4/3} || pseudo {8/3} || pseudo {8/3} || dual {4/3}, proceding in alternating directions each)

o...4/3o...       | 4 * * * | 2 2 0 0 0 0 0 0 0 | 1 2 1 0 0 0 0 0
.o..4/3.o..       | * 8 * * | 0 1 1 1 1 0 0 0 0 | 0 1 1 1 1 0 0 0
..o.4/3..o.       | * * 8 * | 0 0 0 0 1 1 1 1 0 | 0 0 0 1 1 1 1 0
...o4/3...o       | * * * 4 | 0 0 0 0 0 0 0 2 2 | 0 0 0 0 0 1 2 1
------------------+---------+-------------------+----------------
x...   ....       | 2 0 0 0 | 4 * * * * * * * * | 1 1 0 0 0 0 0 0
oo..4/3oo..&#x    | 1 1 0 0 | * 8 * * * * * * * | 0 1 1 0 0 0 0 0
.x..   ....       | 0 2 0 0 | * * 4 * * * * * * | 0 1 0 1 0 0 0 0
....   .x..       | 0 2 0 0 | * * * 4 * * * * * | 0 0 1 0 1 0 0 0
.oo.4/3.oo.&#(-x) | 0 1 1 0 | * * * * 8 * * * * | 0 0 0 1 1 0 0 0
..x.   ....       | 0 0 2 0 | * * * * * 4 * * * | 0 0 0 1 0 1 0 0
....   ..x.       | 0 0 2 0 | * * * * * * 4 * * | 0 0 0 0 1 0 1 0
..oo4/3..oo&#x    | 0 0 1 1 | * * * * * * * 8 * | 0 0 0 0 0 1 1 0
....   ...x       | 0 0 0 2 | * * * * * * * * 4 | 0 0 0 0 0 0 1 1
------------------+---------+-------------------+----------------
x...4/3o...       | 4 0 0 0 | 4 0 0 0 0 0 0 0 0 | 1 * * * * * * *
xx..   ....&#x    | 2 2 0 0 | 1 2 1 0 0 0 0 0 0 | * 4 * * * * * *
....   ox..&#x    | 1 2 0 0 | 0 2 0 1 0 0 0 0 0 | * * 4 * * * * *
.xx.   ....&#(-x) | 0 2 2 0 | 0 0 1 0 2 1 0 0 0 | * * * 4 * * * *
....   .xx.&#(-x) | 0 2 2 0 | 0 0 0 1 2 0 1 0 0 | * * * * 4 * * *
..xo   ....&#x    | 0 0 2 1 | 0 0 0 0 0 1 0 2 0 | * * * * * 4 * *
....   ..xx&#x    | 0 0 2 2 | 0 0 0 0 0 0 1 2 1 | * * * * * * 4 *
...o4/3...x       | 0 0 0 4 | 0 0 0 0 0 0 0 0 4 | * * * * * * * 1

or
o...4/3o...       & | 8  * | 2  2 0 0 0 | 1 2 1 0
.o..4/3.o..       & | * 16 | 0  1 1 1 1 | 0 1 1 2
--------------------+------+------------+--------
x...   ....       & | 2  0 | 8  * * * * | 1 1 0 0
oo..4/3oo..&#x    & | 1  1 | * 16 * * * | 0 1 1 0
.x..   ....       & | 0  2 | *  * 8 * * | 0 1 0 1
....   .x..       & | 0  2 | *  * * 8 * | 0 0 1 1
.oo.4/3.oo.&#(-x)   | 0  2 | *  * * * 8 | 0 0 0 2
--------------------+------+------------+--------
x...4/3o...       & | 4  0 | 4  0 0 0 0 | 2 * * *
xx..   ....&#x    & | 2  2 | 1  2 1 0 0 | * 8 * *
....   ox..&#x    & | 1  2 | 0  2 0 1 0 | * * 8 *
.xx.   ....&#(-x) & | 0  4 | 0  0 1 1 2 | * * * 8